Question 225063
{{{(x^3-3x^2+x-3)/(x^2+1)}}}

<pre><font size = 4 color = "indigo"><b>
There are two methods.

Method 1: Long division:

You must insert +0x<sup>2</sup> in the divisor (denominator):

                        x - 3                   
           ------------------
x<sup>2</sup> + 0x + 1)x<sup>3</sup> - 3x<sup>2</sup> +  x - 3
            x<sup>3</sup> + 0x<sup>2</sup> +  x
            -------------
               - 3x<sup>2</sup> + 0x - 3
               - 3x<sup>2</sup> - 0x - 3
               --------------
                            0

Answer: x - 3

Method 2:  Factoring by grouping:

{{{(x^3-3x^2+x-3)/(x^2+1)}}}

Factor {{{x^2}}} out of the first two terms in the numerator:

{{{(x^2(x-3)+x-3)/(x^2+1)}}}

Factor {{{""+1}}} out of the last two terms of the numerator:

{{{(x^2(x-3)+1(x-3))/(x^2+1)}}}

Factor {{{(x-3)}}} out of the two terms in the numerator:

{{{((x^2+1)(x-3))/(x^2+1)}}}

Cancel the {{{x^2+1}}}'s

{{{(cross((x^2+1))(x-3))/cross((x^2+1))}}}

{{{x-3}}}

Edwin</pre>